# Limit behaviour of a singular perturbation problem for the biharmonic operator

## TL;DR

This paper investigates the asymptotic behavior of solutions to a singular perturbation problem involving the biharmonic operator, revealing convergence to a free boundary problem, monotonicity properties, and solution regularity issues.

## Contribution

It introduces new convergence results, a monotonicity formula, and analyzes solution behavior near the zero level set for a biharmonic singular perturbation problem.

## Key findings

- Solutions converge to a free boundary problem driven by a biharmonic operator.
- A monotonicity formula in the plane is established.
- Counterexamples show potential irregularity without structural assumptions.

## Abstract

We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in a previous paper of ours, and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small values of the perturbation parameter. Some counterexamples to the uniform regularity are also provided if one does not impose some structural assumptions on the forcing term.